# Properties

 Label 8470.2.a.dc.1.1 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.19898000.1 Defining polynomial: $$x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.86564$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.86564 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.86564 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.21187 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.86564 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.86564 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.21187 q^{9} +1.00000 q^{10} -2.86564 q^{12} +5.63670 q^{13} -1.00000 q^{14} -2.86564 q^{15} +1.00000 q^{16} +2.32986 q^{17} +5.21187 q^{18} +7.81494 q^{19} +1.00000 q^{20} +2.86564 q^{21} +7.98434 q^{23} -2.86564 q^{24} +1.00000 q^{25} +5.63670 q^{26} -6.33840 q^{27} -1.00000 q^{28} -2.99446 q^{29} -2.86564 q^{30} +5.08533 q^{31} +1.00000 q^{32} +2.32986 q^{34} -1.00000 q^{35} +5.21187 q^{36} -1.38681 q^{37} +7.81494 q^{38} -16.1527 q^{39} +1.00000 q^{40} -6.90154 q^{41} +2.86564 q^{42} +11.5492 q^{43} +5.21187 q^{45} +7.98434 q^{46} -5.14520 q^{47} -2.86564 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.67652 q^{51} +5.63670 q^{52} +0.582695 q^{53} -6.33840 q^{54} -1.00000 q^{56} -22.3948 q^{57} -2.99446 q^{58} +13.8292 q^{59} -2.86564 q^{60} -7.82443 q^{61} +5.08533 q^{62} -5.21187 q^{63} +1.00000 q^{64} +5.63670 q^{65} -0.0485769 q^{67} +2.32986 q^{68} -22.8802 q^{69} -1.00000 q^{70} +4.45361 q^{71} +5.21187 q^{72} +0.336814 q^{73} -1.38681 q^{74} -2.86564 q^{75} +7.81494 q^{76} -16.1527 q^{78} -1.05291 q^{79} +1.00000 q^{80} +2.52795 q^{81} -6.90154 q^{82} -10.8315 q^{83} +2.86564 q^{84} +2.32986 q^{85} +11.5492 q^{86} +8.58103 q^{87} -18.5268 q^{89} +5.21187 q^{90} -5.63670 q^{91} +7.98434 q^{92} -14.5727 q^{93} -5.14520 q^{94} +7.81494 q^{95} -2.86564 q^{96} -5.33180 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} + O(q^{10})$$ $$6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} + 6 q^{10} - q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 9 q^{17} + 3 q^{18} + 12 q^{19} + 6 q^{20} + q^{21} + 4 q^{23} - q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} - 6 q^{28} + 15 q^{29} - q^{30} + 8 q^{31} + 6 q^{32} + 9 q^{34} - 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} - 19 q^{39} + 6 q^{40} + 4 q^{41} + q^{42} + 30 q^{43} + 3 q^{45} + 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 16 q^{51} + 9 q^{52} - 6 q^{53} - 4 q^{54} - 6 q^{56} - 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} - 14 q^{61} + 8 q^{62} - 3 q^{63} + 6 q^{64} + 9 q^{65} + 18 q^{67} + 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} + 3 q^{72} + 23 q^{73} - 4 q^{74} - q^{75} + 12 q^{76} - 19 q^{78} + 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} + 25 q^{83} + q^{84} + 9 q^{85} + 30 q^{86} + 14 q^{87} - 18 q^{89} + 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} - 7 q^{94} + 12 q^{95} - q^{96} + 7 q^{97} + 6 q^{98} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −2.86564 −1.65448 −0.827238 0.561852i $$-0.810089\pi$$
−0.827238 + 0.561852i $$0.810089\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −2.86564 −1.16989
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 5.21187 1.73729
$$10$$ 1.00000 0.316228
$$11$$ 0 0
$$12$$ −2.86564 −0.827238
$$13$$ 5.63670 1.56334 0.781669 0.623693i $$-0.214369\pi$$
0.781669 + 0.623693i $$0.214369\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ −2.86564 −0.739904
$$16$$ 1.00000 0.250000
$$17$$ 2.32986 0.565074 0.282537 0.959256i $$-0.408824\pi$$
0.282537 + 0.959256i $$0.408824\pi$$
$$18$$ 5.21187 1.22845
$$19$$ 7.81494 1.79287 0.896435 0.443175i $$-0.146148\pi$$
0.896435 + 0.443175i $$0.146148\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.86564 0.625333
$$22$$ 0 0
$$23$$ 7.98434 1.66485 0.832425 0.554138i $$-0.186952\pi$$
0.832425 + 0.554138i $$0.186952\pi$$
$$24$$ −2.86564 −0.584945
$$25$$ 1.00000 0.200000
$$26$$ 5.63670 1.10545
$$27$$ −6.33840 −1.21983
$$28$$ −1.00000 −0.188982
$$29$$ −2.99446 −0.556057 −0.278029 0.960573i $$-0.589681\pi$$
−0.278029 + 0.960573i $$0.589681\pi$$
$$30$$ −2.86564 −0.523191
$$31$$ 5.08533 0.913352 0.456676 0.889633i $$-0.349040\pi$$
0.456676 + 0.889633i $$0.349040\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 2.32986 0.399567
$$35$$ −1.00000 −0.169031
$$36$$ 5.21187 0.868644
$$37$$ −1.38681 −0.227989 −0.113995 0.993481i $$-0.536365\pi$$
−0.113995 + 0.993481i $$0.536365\pi$$
$$38$$ 7.81494 1.26775
$$39$$ −16.1527 −2.58650
$$40$$ 1.00000 0.158114
$$41$$ −6.90154 −1.07784 −0.538920 0.842357i $$-0.681167\pi$$
−0.538920 + 0.842357i $$0.681167\pi$$
$$42$$ 2.86564 0.442177
$$43$$ 11.5492 1.76124 0.880622 0.473820i $$-0.157125\pi$$
0.880622 + 0.473820i $$0.157125\pi$$
$$44$$ 0 0
$$45$$ 5.21187 0.776939
$$46$$ 7.98434 1.17723
$$47$$ −5.14520 −0.750504 −0.375252 0.926923i $$-0.622444\pi$$
−0.375252 + 0.926923i $$0.622444\pi$$
$$48$$ −2.86564 −0.413619
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ −6.67652 −0.934900
$$52$$ 5.63670 0.781669
$$53$$ 0.582695 0.0800393 0.0400196 0.999199i $$-0.487258\pi$$
0.0400196 + 0.999199i $$0.487258\pi$$
$$54$$ −6.33840 −0.862547
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −22.3948 −2.96626
$$58$$ −2.99446 −0.393192
$$59$$ 13.8292 1.80041 0.900204 0.435468i $$-0.143417\pi$$
0.900204 + 0.435468i $$0.143417\pi$$
$$60$$ −2.86564 −0.369952
$$61$$ −7.82443 −1.00182 −0.500908 0.865501i $$-0.667000\pi$$
−0.500908 + 0.865501i $$0.667000\pi$$
$$62$$ 5.08533 0.645838
$$63$$ −5.21187 −0.656633
$$64$$ 1.00000 0.125000
$$65$$ 5.63670 0.699146
$$66$$ 0 0
$$67$$ −0.0485769 −0.00593462 −0.00296731 0.999996i $$-0.500945\pi$$
−0.00296731 + 0.999996i $$0.500945\pi$$
$$68$$ 2.32986 0.282537
$$69$$ −22.8802 −2.75445
$$70$$ −1.00000 −0.119523
$$71$$ 4.45361 0.528546 0.264273 0.964448i $$-0.414868\pi$$
0.264273 + 0.964448i $$0.414868\pi$$
$$72$$ 5.21187 0.614224
$$73$$ 0.336814 0.0394210 0.0197105 0.999806i $$-0.493726\pi$$
0.0197105 + 0.999806i $$0.493726\pi$$
$$74$$ −1.38681 −0.161213
$$75$$ −2.86564 −0.330895
$$76$$ 7.81494 0.896435
$$77$$ 0 0
$$78$$ −16.1527 −1.82893
$$79$$ −1.05291 −0.118462 −0.0592308 0.998244i $$-0.518865\pi$$
−0.0592308 + 0.998244i $$0.518865\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 2.52795 0.280883
$$82$$ −6.90154 −0.762148
$$83$$ −10.8315 −1.18891 −0.594455 0.804129i $$-0.702632\pi$$
−0.594455 + 0.804129i $$0.702632\pi$$
$$84$$ 2.86564 0.312666
$$85$$ 2.32986 0.252709
$$86$$ 11.5492 1.24539
$$87$$ 8.58103 0.919983
$$88$$ 0 0
$$89$$ −18.5268 −1.96384 −0.981921 0.189291i $$-0.939381\pi$$
−0.981921 + 0.189291i $$0.939381\pi$$
$$90$$ 5.21187 0.549379
$$91$$ −5.63670 −0.590886
$$92$$ 7.98434 0.832425
$$93$$ −14.5727 −1.51112
$$94$$ −5.14520 −0.530686
$$95$$ 7.81494 0.801796
$$96$$ −2.86564 −0.292473
$$97$$ −5.33180 −0.541362 −0.270681 0.962669i $$-0.587249\pi$$
−0.270681 + 0.962669i $$0.587249\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 18.2701 1.81795 0.908973 0.416854i $$-0.136867\pi$$
0.908973 + 0.416854i $$0.136867\pi$$
$$102$$ −6.67652 −0.661074
$$103$$ −11.2347 −1.10699 −0.553495 0.832852i $$-0.686706\pi$$
−0.553495 + 0.832852i $$0.686706\pi$$
$$104$$ 5.63670 0.552723
$$105$$ 2.86564 0.279657
$$106$$ 0.582695 0.0565963
$$107$$ −4.89157 −0.472886 −0.236443 0.971645i $$-0.575982\pi$$
−0.236443 + 0.971645i $$0.575982\pi$$
$$108$$ −6.33840 −0.609913
$$109$$ −6.26442 −0.600023 −0.300011 0.953936i $$-0.596991\pi$$
−0.300011 + 0.953936i $$0.596991\pi$$
$$110$$ 0 0
$$111$$ 3.97408 0.377203
$$112$$ −1.00000 −0.0944911
$$113$$ 10.1651 0.956251 0.478126 0.878291i $$-0.341316\pi$$
0.478126 + 0.878291i $$0.341316\pi$$
$$114$$ −22.3948 −2.09746
$$115$$ 7.98434 0.744544
$$116$$ −2.99446 −0.278029
$$117$$ 29.3777 2.71597
$$118$$ 13.8292 1.27308
$$119$$ −2.32986 −0.213578
$$120$$ −2.86564 −0.261596
$$121$$ 0 0
$$122$$ −7.82443 −0.708391
$$123$$ 19.7773 1.78326
$$124$$ 5.08533 0.456676
$$125$$ 1.00000 0.0894427
$$126$$ −5.21187 −0.464310
$$127$$ −6.58727 −0.584526 −0.292263 0.956338i $$-0.594408\pi$$
−0.292263 + 0.956338i $$0.594408\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −33.0959 −2.91393
$$130$$ 5.63670 0.494371
$$131$$ −1.20821 −0.105562 −0.0527808 0.998606i $$-0.516808\pi$$
−0.0527808 + 0.998606i $$0.516808\pi$$
$$132$$ 0 0
$$133$$ −7.81494 −0.677641
$$134$$ −0.0485769 −0.00419641
$$135$$ −6.33840 −0.545523
$$136$$ 2.32986 0.199784
$$137$$ 14.0829 1.20318 0.601590 0.798805i $$-0.294534\pi$$
0.601590 + 0.798805i $$0.294534\pi$$
$$138$$ −22.8802 −1.94769
$$139$$ −2.74290 −0.232650 −0.116325 0.993211i $$-0.537111\pi$$
−0.116325 + 0.993211i $$0.537111\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 14.7443 1.24169
$$142$$ 4.45361 0.373739
$$143$$ 0 0
$$144$$ 5.21187 0.434322
$$145$$ −2.99446 −0.248676
$$146$$ 0.336814 0.0278749
$$147$$ −2.86564 −0.236354
$$148$$ −1.38681 −0.113995
$$149$$ 17.1642 1.40615 0.703075 0.711116i $$-0.251810\pi$$
0.703075 + 0.711116i $$0.251810\pi$$
$$150$$ −2.86564 −0.233978
$$151$$ −5.94304 −0.483637 −0.241819 0.970321i $$-0.577744\pi$$
−0.241819 + 0.970321i $$0.577744\pi$$
$$152$$ 7.81494 0.633875
$$153$$ 12.1429 0.981696
$$154$$ 0 0
$$155$$ 5.08533 0.408464
$$156$$ −16.1527 −1.29325
$$157$$ −16.7947 −1.34037 −0.670183 0.742196i $$-0.733784\pi$$
−0.670183 + 0.742196i $$0.733784\pi$$
$$158$$ −1.05291 −0.0837650
$$159$$ −1.66979 −0.132423
$$160$$ 1.00000 0.0790569
$$161$$ −7.98434 −0.629254
$$162$$ 2.52795 0.198614
$$163$$ 2.43041 0.190364 0.0951821 0.995460i $$-0.469657\pi$$
0.0951821 + 0.995460i $$0.469657\pi$$
$$164$$ −6.90154 −0.538920
$$165$$ 0 0
$$166$$ −10.8315 −0.840686
$$167$$ 11.0306 0.853572 0.426786 0.904353i $$-0.359646\pi$$
0.426786 + 0.904353i $$0.359646\pi$$
$$168$$ 2.86564 0.221089
$$169$$ 18.7723 1.44403
$$170$$ 2.32986 0.178692
$$171$$ 40.7304 3.11473
$$172$$ 11.5492 0.880622
$$173$$ −14.6947 −1.11721 −0.558607 0.829433i $$-0.688664\pi$$
−0.558607 + 0.829433i $$0.688664\pi$$
$$174$$ 8.58103 0.650526
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −39.6294 −2.97873
$$178$$ −18.5268 −1.38865
$$179$$ 4.51392 0.337387 0.168693 0.985669i $$-0.446045\pi$$
0.168693 + 0.985669i $$0.446045\pi$$
$$180$$ 5.21187 0.388470
$$181$$ −10.1770 −0.756451 −0.378225 0.925714i $$-0.623466\pi$$
−0.378225 + 0.925714i $$0.623466\pi$$
$$182$$ −5.63670 −0.417820
$$183$$ 22.4220 1.65748
$$184$$ 7.98434 0.588613
$$185$$ −1.38681 −0.101960
$$186$$ −14.5727 −1.06852
$$187$$ 0 0
$$188$$ −5.14520 −0.375252
$$189$$ 6.33840 0.461051
$$190$$ 7.81494 0.566955
$$191$$ 11.9914 0.867669 0.433834 0.900993i $$-0.357160\pi$$
0.433834 + 0.900993i $$0.357160\pi$$
$$192$$ −2.86564 −0.206809
$$193$$ 17.8748 1.28666 0.643330 0.765589i $$-0.277552\pi$$
0.643330 + 0.765589i $$0.277552\pi$$
$$194$$ −5.33180 −0.382801
$$195$$ −16.1527 −1.15672
$$196$$ 1.00000 0.0714286
$$197$$ 11.7871 0.839795 0.419897 0.907572i $$-0.362066\pi$$
0.419897 + 0.907572i $$0.362066\pi$$
$$198$$ 0 0
$$199$$ −13.9342 −0.987767 −0.493884 0.869528i $$-0.664423\pi$$
−0.493884 + 0.869528i $$0.664423\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0.139204 0.00981868
$$202$$ 18.2701 1.28548
$$203$$ 2.99446 0.210170
$$204$$ −6.67652 −0.467450
$$205$$ −6.90154 −0.482025
$$206$$ −11.2347 −0.782760
$$207$$ 41.6133 2.89233
$$208$$ 5.63670 0.390835
$$209$$ 0 0
$$210$$ 2.86564 0.197748
$$211$$ −26.4497 −1.82087 −0.910435 0.413653i $$-0.864253\pi$$
−0.910435 + 0.413653i $$0.864253\pi$$
$$212$$ 0.582695 0.0400196
$$213$$ −12.7624 −0.874467
$$214$$ −4.89157 −0.334381
$$215$$ 11.5492 0.787652
$$216$$ −6.33840 −0.431274
$$217$$ −5.08533 −0.345215
$$218$$ −6.26442 −0.424280
$$219$$ −0.965185 −0.0652211
$$220$$ 0 0
$$221$$ 13.1327 0.883401
$$222$$ 3.97408 0.266723
$$223$$ 24.5893 1.64662 0.823311 0.567591i $$-0.192124\pi$$
0.823311 + 0.567591i $$0.192124\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 5.21187 0.347458
$$226$$ 10.1651 0.676172
$$227$$ 5.59242 0.371182 0.185591 0.982627i $$-0.440580\pi$$
0.185591 + 0.982627i $$0.440580\pi$$
$$228$$ −22.3948 −1.48313
$$229$$ −8.10340 −0.535488 −0.267744 0.963490i $$-0.586278\pi$$
−0.267744 + 0.963490i $$0.586278\pi$$
$$230$$ 7.98434 0.526472
$$231$$ 0 0
$$232$$ −2.99446 −0.196596
$$233$$ −11.9807 −0.784882 −0.392441 0.919777i $$-0.628369\pi$$
−0.392441 + 0.919777i $$0.628369\pi$$
$$234$$ 29.3777 1.92048
$$235$$ −5.14520 −0.335636
$$236$$ 13.8292 0.900204
$$237$$ 3.01726 0.195992
$$238$$ −2.32986 −0.151022
$$239$$ 6.46612 0.418258 0.209129 0.977888i $$-0.432937\pi$$
0.209129 + 0.977888i $$0.432937\pi$$
$$240$$ −2.86564 −0.184976
$$241$$ −20.3441 −1.31048 −0.655238 0.755423i $$-0.727432\pi$$
−0.655238 + 0.755423i $$0.727432\pi$$
$$242$$ 0 0
$$243$$ 11.7710 0.755112
$$244$$ −7.82443 −0.500908
$$245$$ 1.00000 0.0638877
$$246$$ 19.7773 1.26096
$$247$$ 44.0504 2.80286
$$248$$ 5.08533 0.322919
$$249$$ 31.0391 1.96702
$$250$$ 1.00000 0.0632456
$$251$$ 12.5842 0.794311 0.397155 0.917751i $$-0.369997\pi$$
0.397155 + 0.917751i $$0.369997\pi$$
$$252$$ −5.21187 −0.328317
$$253$$ 0 0
$$254$$ −6.58727 −0.413322
$$255$$ −6.67652 −0.418100
$$256$$ 1.00000 0.0625000
$$257$$ 3.45995 0.215826 0.107913 0.994160i $$-0.465583\pi$$
0.107913 + 0.994160i $$0.465583\pi$$
$$258$$ −33.0959 −2.06046
$$259$$ 1.38681 0.0861719
$$260$$ 5.63670 0.349573
$$261$$ −15.6067 −0.966032
$$262$$ −1.20821 −0.0746434
$$263$$ 8.01789 0.494404 0.247202 0.968964i $$-0.420489\pi$$
0.247202 + 0.968964i $$0.420489\pi$$
$$264$$ 0 0
$$265$$ 0.582695 0.0357946
$$266$$ −7.81494 −0.479165
$$267$$ 53.0912 3.24913
$$268$$ −0.0485769 −0.00296731
$$269$$ −18.5650 −1.13193 −0.565964 0.824430i $$-0.691496\pi$$
−0.565964 + 0.824430i $$0.691496\pi$$
$$270$$ −6.33840 −0.385743
$$271$$ −25.2051 −1.53110 −0.765552 0.643375i $$-0.777534\pi$$
−0.765552 + 0.643375i $$0.777534\pi$$
$$272$$ 2.32986 0.141268
$$273$$ 16.1527 0.977607
$$274$$ 14.0829 0.850777
$$275$$ 0 0
$$276$$ −22.8802 −1.37723
$$277$$ 12.2281 0.734714 0.367357 0.930080i $$-0.380263\pi$$
0.367357 + 0.930080i $$0.380263\pi$$
$$278$$ −2.74290 −0.164508
$$279$$ 26.5041 1.58676
$$280$$ −1.00000 −0.0597614
$$281$$ −26.5934 −1.58643 −0.793216 0.608941i $$-0.791595\pi$$
−0.793216 + 0.608941i $$0.791595\pi$$
$$282$$ 14.7443 0.878008
$$283$$ 13.7220 0.815689 0.407845 0.913051i $$-0.366281\pi$$
0.407845 + 0.913051i $$0.366281\pi$$
$$284$$ 4.45361 0.264273
$$285$$ −22.3948 −1.32655
$$286$$ 0 0
$$287$$ 6.90154 0.407385
$$288$$ 5.21187 0.307112
$$289$$ −11.5718 −0.680692
$$290$$ −2.99446 −0.175841
$$291$$ 15.2790 0.895670
$$292$$ 0.336814 0.0197105
$$293$$ −0.714463 −0.0417394 −0.0208697 0.999782i $$-0.506644\pi$$
−0.0208697 + 0.999782i $$0.506644\pi$$
$$294$$ −2.86564 −0.167127
$$295$$ 13.8292 0.805167
$$296$$ −1.38681 −0.0806064
$$297$$ 0 0
$$298$$ 17.1642 0.994298
$$299$$ 45.0053 2.60272
$$300$$ −2.86564 −0.165448
$$301$$ −11.5492 −0.665688
$$302$$ −5.94304 −0.341983
$$303$$ −52.3556 −3.00775
$$304$$ 7.81494 0.448218
$$305$$ −7.82443 −0.448026
$$306$$ 12.1429 0.694164
$$307$$ 19.3442 1.10403 0.552015 0.833834i $$-0.313859\pi$$
0.552015 + 0.833834i $$0.313859\pi$$
$$308$$ 0 0
$$309$$ 32.1946 1.83149
$$310$$ 5.08533 0.288827
$$311$$ 1.82729 0.103616 0.0518080 0.998657i $$-0.483502\pi$$
0.0518080 + 0.998657i $$0.483502\pi$$
$$312$$ −16.1527 −0.914467
$$313$$ −9.83108 −0.555686 −0.277843 0.960627i $$-0.589619\pi$$
−0.277843 + 0.960627i $$0.589619\pi$$
$$314$$ −16.7947 −0.947782
$$315$$ −5.21187 −0.293655
$$316$$ −1.05291 −0.0592308
$$317$$ −13.8545 −0.778146 −0.389073 0.921207i $$-0.627205\pi$$
−0.389073 + 0.921207i $$0.627205\pi$$
$$318$$ −1.66979 −0.0936372
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 14.0175 0.782378
$$322$$ −7.98434 −0.444950
$$323$$ 18.2077 1.01310
$$324$$ 2.52795 0.140442
$$325$$ 5.63670 0.312668
$$326$$ 2.43041 0.134608
$$327$$ 17.9516 0.992723
$$328$$ −6.90154 −0.381074
$$329$$ 5.14520 0.283664
$$330$$ 0 0
$$331$$ 8.02084 0.440865 0.220433 0.975402i $$-0.429253\pi$$
0.220433 + 0.975402i $$0.429253\pi$$
$$332$$ −10.8315 −0.594455
$$333$$ −7.22784 −0.396083
$$334$$ 11.0306 0.603567
$$335$$ −0.0485769 −0.00265404
$$336$$ 2.86564 0.156333
$$337$$ 11.6060 0.632218 0.316109 0.948723i $$-0.397624\pi$$
0.316109 + 0.948723i $$0.397624\pi$$
$$338$$ 18.7723 1.02108
$$339$$ −29.1294 −1.58209
$$340$$ 2.32986 0.126354
$$341$$ 0 0
$$342$$ 40.7304 2.20245
$$343$$ −1.00000 −0.0539949
$$344$$ 11.5492 0.622694
$$345$$ −22.8802 −1.23183
$$346$$ −14.6947 −0.789989
$$347$$ 26.4301 1.41884 0.709420 0.704786i $$-0.248957\pi$$
0.709420 + 0.704786i $$0.248957\pi$$
$$348$$ 8.58103 0.459991
$$349$$ −27.3777 −1.46549 −0.732747 0.680501i $$-0.761762\pi$$
−0.732747 + 0.680501i $$0.761762\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ −35.7276 −1.90700
$$352$$ 0 0
$$353$$ 21.3772 1.13779 0.568897 0.822409i $$-0.307370\pi$$
0.568897 + 0.822409i $$0.307370\pi$$
$$354$$ −39.6294 −2.10628
$$355$$ 4.45361 0.236373
$$356$$ −18.5268 −0.981921
$$357$$ 6.67652 0.353359
$$358$$ 4.51392 0.238568
$$359$$ 10.8904 0.574774 0.287387 0.957815i $$-0.407213\pi$$
0.287387 + 0.957815i $$0.407213\pi$$
$$360$$ 5.21187 0.274689
$$361$$ 42.0733 2.21439
$$362$$ −10.1770 −0.534891
$$363$$ 0 0
$$364$$ −5.63670 −0.295443
$$365$$ 0.336814 0.0176296
$$366$$ 22.4220 1.17201
$$367$$ 11.3108 0.590416 0.295208 0.955433i $$-0.404611\pi$$
0.295208 + 0.955433i $$0.404611\pi$$
$$368$$ 7.98434 0.416213
$$369$$ −35.9699 −1.87252
$$370$$ −1.38681 −0.0720966
$$371$$ −0.582695 −0.0302520
$$372$$ −14.5727 −0.755559
$$373$$ −11.6658 −0.604034 −0.302017 0.953303i $$-0.597660\pi$$
−0.302017 + 0.953303i $$0.597660\pi$$
$$374$$ 0 0
$$375$$ −2.86564 −0.147981
$$376$$ −5.14520 −0.265343
$$377$$ −16.8789 −0.869305
$$378$$ 6.33840 0.326012
$$379$$ 0.617088 0.0316977 0.0158488 0.999874i $$-0.494955\pi$$
0.0158488 + 0.999874i $$0.494955\pi$$
$$380$$ 7.81494 0.400898
$$381$$ 18.8767 0.967084
$$382$$ 11.9914 0.613535
$$383$$ −26.4471 −1.35138 −0.675692 0.737184i $$-0.736155\pi$$
−0.675692 + 0.737184i $$0.736155\pi$$
$$384$$ −2.86564 −0.146236
$$385$$ 0 0
$$386$$ 17.8748 0.909806
$$387$$ 60.1931 3.05979
$$388$$ −5.33180 −0.270681
$$389$$ 1.60446 0.0813492 0.0406746 0.999172i $$-0.487049\pi$$
0.0406746 + 0.999172i $$0.487049\pi$$
$$390$$ −16.1527 −0.817924
$$391$$ 18.6024 0.940763
$$392$$ 1.00000 0.0505076
$$393$$ 3.46228 0.174649
$$394$$ 11.7871 0.593825
$$395$$ −1.05291 −0.0529776
$$396$$ 0 0
$$397$$ 5.28819 0.265407 0.132703 0.991156i $$-0.457634\pi$$
0.132703 + 0.991156i $$0.457634\pi$$
$$398$$ −13.9342 −0.698457
$$399$$ 22.3948 1.12114
$$400$$ 1.00000 0.0500000
$$401$$ −13.6059 −0.679446 −0.339723 0.940526i $$-0.610333\pi$$
−0.339723 + 0.940526i $$0.610333\pi$$
$$402$$ 0.139204 0.00694285
$$403$$ 28.6645 1.42788
$$404$$ 18.2701 0.908973
$$405$$ 2.52795 0.125615
$$406$$ 2.99446 0.148613
$$407$$ 0 0
$$408$$ −6.67652 −0.330537
$$409$$ 3.40230 0.168233 0.0841164 0.996456i $$-0.473193\pi$$
0.0841164 + 0.996456i $$0.473193\pi$$
$$410$$ −6.90154 −0.340843
$$411$$ −40.3564 −1.99063
$$412$$ −11.2347 −0.553495
$$413$$ −13.8292 −0.680490
$$414$$ 41.6133 2.04518
$$415$$ −10.8315 −0.531697
$$416$$ 5.63670 0.276362
$$417$$ 7.86016 0.384914
$$418$$ 0 0
$$419$$ −39.5263 −1.93099 −0.965493 0.260428i $$-0.916136\pi$$
−0.965493 + 0.260428i $$0.916136\pi$$
$$420$$ 2.86564 0.139829
$$421$$ −15.6617 −0.763304 −0.381652 0.924306i $$-0.624645\pi$$
−0.381652 + 0.924306i $$0.624645\pi$$
$$422$$ −26.4497 −1.28755
$$423$$ −26.8161 −1.30384
$$424$$ 0.582695 0.0282982
$$425$$ 2.32986 0.113015
$$426$$ −12.7624 −0.618341
$$427$$ 7.82443 0.378651
$$428$$ −4.89157 −0.236443
$$429$$ 0 0
$$430$$ 11.5492 0.556954
$$431$$ −5.63150 −0.271260 −0.135630 0.990760i $$-0.543306\pi$$
−0.135630 + 0.990760i $$0.543306\pi$$
$$432$$ −6.33840 −0.304956
$$433$$ −34.5223 −1.65903 −0.829517 0.558481i $$-0.811384\pi$$
−0.829517 + 0.558481i $$0.811384\pi$$
$$434$$ −5.08533 −0.244104
$$435$$ 8.58103 0.411429
$$436$$ −6.26442 −0.300011
$$437$$ 62.3972 2.98486
$$438$$ −0.965185 −0.0461183
$$439$$ −30.0533 −1.43437 −0.717184 0.696884i $$-0.754569\pi$$
−0.717184 + 0.696884i $$0.754569\pi$$
$$440$$ 0 0
$$441$$ 5.21187 0.248184
$$442$$ 13.1327 0.624659
$$443$$ −32.0651 −1.52346 −0.761730 0.647894i $$-0.775650\pi$$
−0.761730 + 0.647894i $$0.775650\pi$$
$$444$$ 3.97408 0.188601
$$445$$ −18.5268 −0.878257
$$446$$ 24.5893 1.16434
$$447$$ −49.1865 −2.32644
$$448$$ −1.00000 −0.0472456
$$449$$ −9.27706 −0.437811 −0.218906 0.975746i $$-0.570249\pi$$
−0.218906 + 0.975746i $$0.570249\pi$$
$$450$$ 5.21187 0.245690
$$451$$ 0 0
$$452$$ 10.1651 0.478126
$$453$$ 17.0306 0.800166
$$454$$ 5.59242 0.262465
$$455$$ −5.63670 −0.264252
$$456$$ −22.3948 −1.04873
$$457$$ 20.0782 0.939217 0.469608 0.882875i $$-0.344395\pi$$
0.469608 + 0.882875i $$0.344395\pi$$
$$458$$ −8.10340 −0.378647
$$459$$ −14.7676 −0.689291
$$460$$ 7.98434 0.372272
$$461$$ −14.6490 −0.682272 −0.341136 0.940014i $$-0.610812\pi$$
−0.341136 + 0.940014i $$0.610812\pi$$
$$462$$ 0 0
$$463$$ 11.0973 0.515737 0.257869 0.966180i $$-0.416980\pi$$
0.257869 + 0.966180i $$0.416980\pi$$
$$464$$ −2.99446 −0.139014
$$465$$ −14.5727 −0.675793
$$466$$ −11.9807 −0.554995
$$467$$ 24.6923 1.14262 0.571311 0.820734i $$-0.306435\pi$$
0.571311 + 0.820734i $$0.306435\pi$$
$$468$$ 29.3777 1.35798
$$469$$ 0.0485769 0.00224307
$$470$$ −5.14520 −0.237330
$$471$$ 48.1276 2.21760
$$472$$ 13.8292 0.636540
$$473$$ 0 0
$$474$$ 3.01726 0.138587
$$475$$ 7.81494 0.358574
$$476$$ −2.32986 −0.106789
$$477$$ 3.03693 0.139051
$$478$$ 6.46612 0.295753
$$479$$ 31.2114 1.42608 0.713042 0.701121i $$-0.247317\pi$$
0.713042 + 0.701121i $$0.247317\pi$$
$$480$$ −2.86564 −0.130798
$$481$$ −7.81700 −0.356424
$$482$$ −20.3441 −0.926646
$$483$$ 22.8802 1.04109
$$484$$ 0 0
$$485$$ −5.33180 −0.242104
$$486$$ 11.7710 0.533945
$$487$$ 3.02777 0.137201 0.0686007 0.997644i $$-0.478147\pi$$
0.0686007 + 0.997644i $$0.478147\pi$$
$$488$$ −7.82443 −0.354195
$$489$$ −6.96466 −0.314953
$$490$$ 1.00000 0.0451754
$$491$$ 14.8069 0.668228 0.334114 0.942533i $$-0.391563\pi$$
0.334114 + 0.942533i $$0.391563\pi$$
$$492$$ 19.7773 0.891630
$$493$$ −6.97666 −0.314213
$$494$$ 44.0504 1.98192
$$495$$ 0 0
$$496$$ 5.08533 0.228338
$$497$$ −4.45361 −0.199772
$$498$$ 31.0391 1.39089
$$499$$ −6.99445 −0.313115 −0.156557 0.987669i $$-0.550040\pi$$
−0.156557 + 0.987669i $$0.550040\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −31.6096 −1.41221
$$502$$ 12.5842 0.561663
$$503$$ 1.93800 0.0864112 0.0432056 0.999066i $$-0.486243\pi$$
0.0432056 + 0.999066i $$0.486243\pi$$
$$504$$ −5.21187 −0.232155
$$505$$ 18.2701 0.813011
$$506$$ 0 0
$$507$$ −53.7947 −2.38910
$$508$$ −6.58727 −0.292263
$$509$$ 12.9518 0.574081 0.287040 0.957919i $$-0.407329\pi$$
0.287040 + 0.957919i $$0.407329\pi$$
$$510$$ −6.67652 −0.295641
$$511$$ −0.336814 −0.0148998
$$512$$ 1.00000 0.0441942
$$513$$ −49.5342 −2.18699
$$514$$ 3.45995 0.152612
$$515$$ −11.2347 −0.495061
$$516$$ −33.0959 −1.45697
$$517$$ 0 0
$$518$$ 1.38681 0.0609327
$$519$$ 42.1095 1.84840
$$520$$ 5.63670 0.247185
$$521$$ −5.90459 −0.258685 −0.129342 0.991600i $$-0.541287\pi$$
−0.129342 + 0.991600i $$0.541287\pi$$
$$522$$ −15.6067 −0.683088
$$523$$ 18.1944 0.795587 0.397794 0.917475i $$-0.369776\pi$$
0.397794 + 0.917475i $$0.369776\pi$$
$$524$$ −1.20821 −0.0527808
$$525$$ 2.86564 0.125067
$$526$$ 8.01789 0.349597
$$527$$ 11.8481 0.516111
$$528$$ 0 0
$$529$$ 40.7497 1.77173
$$530$$ 0.582695 0.0253106
$$531$$ 72.0759 3.12783
$$532$$ −7.81494 −0.338821
$$533$$ −38.9019 −1.68503
$$534$$ 53.0912 2.29748
$$535$$ −4.89157 −0.211481
$$536$$ −0.0485769 −0.00209820
$$537$$ −12.9353 −0.558198
$$538$$ −18.5650 −0.800393
$$539$$ 0 0
$$540$$ −6.33840 −0.272761
$$541$$ 15.1525 0.651457 0.325729 0.945463i $$-0.394390\pi$$
0.325729 + 0.945463i $$0.394390\pi$$
$$542$$ −25.2051 −1.08265
$$543$$ 29.1636 1.25153
$$544$$ 2.32986 0.0998918
$$545$$ −6.26442 −0.268338
$$546$$ 16.1527 0.691272
$$547$$ 10.3369 0.441975 0.220988 0.975277i $$-0.429072\pi$$
0.220988 + 0.975277i $$0.429072\pi$$
$$548$$ 14.0829 0.601590
$$549$$ −40.7799 −1.74044
$$550$$ 0 0
$$551$$ −23.4015 −0.996938
$$552$$ −22.8802 −0.973846
$$553$$ 1.05291 0.0447743
$$554$$ 12.2281 0.519521
$$555$$ 3.97408 0.168690
$$556$$ −2.74290 −0.116325
$$557$$ 6.41056 0.271624 0.135812 0.990735i $$-0.456636\pi$$
0.135812 + 0.990735i $$0.456636\pi$$
$$558$$ 26.5041 1.12201
$$559$$ 65.0996 2.75342
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ −26.5934 −1.12178
$$563$$ 36.3501 1.53197 0.765986 0.642857i $$-0.222251\pi$$
0.765986 + 0.642857i $$0.222251\pi$$
$$564$$ 14.7443 0.620845
$$565$$ 10.1651 0.427649
$$566$$ 13.7220 0.576779
$$567$$ −2.52795 −0.106164
$$568$$ 4.45361 0.186869
$$569$$ 45.1901 1.89447 0.947234 0.320544i $$-0.103866\pi$$
0.947234 + 0.320544i $$0.103866\pi$$
$$570$$ −22.3948 −0.938014
$$571$$ −24.7479 −1.03567 −0.517834 0.855481i $$-0.673262\pi$$
−0.517834 + 0.855481i $$0.673262\pi$$
$$572$$ 0 0
$$573$$ −34.3630 −1.43554
$$574$$ 6.90154 0.288065
$$575$$ 7.98434 0.332970
$$576$$ 5.21187 0.217161
$$577$$ −2.62695 −0.109361 −0.0546807 0.998504i $$-0.517414\pi$$
−0.0546807 + 0.998504i $$0.517414\pi$$
$$578$$ −11.5718 −0.481322
$$579$$ −51.2228 −2.12875
$$580$$ −2.99446 −0.124338
$$581$$ 10.8315 0.449366
$$582$$ 15.2790 0.633334
$$583$$ 0 0
$$584$$ 0.336814 0.0139374
$$585$$ 29.3777 1.21462
$$586$$ −0.714463 −0.0295142
$$587$$ 12.8971 0.532319 0.266159 0.963929i $$-0.414245\pi$$
0.266159 + 0.963929i $$0.414245\pi$$
$$588$$ −2.86564 −0.118177
$$589$$ 39.7416 1.63752
$$590$$ 13.8292 0.569339
$$591$$ −33.7775 −1.38942
$$592$$ −1.38681 −0.0569973
$$593$$ 33.4157 1.37222 0.686110 0.727498i $$-0.259317\pi$$
0.686110 + 0.727498i $$0.259317\pi$$
$$594$$ 0 0
$$595$$ −2.32986 −0.0955149
$$596$$ 17.1642 0.703075
$$597$$ 39.9303 1.63424
$$598$$ 45.0053 1.84040
$$599$$ 34.4823 1.40891 0.704455 0.709749i $$-0.251192\pi$$
0.704455 + 0.709749i $$0.251192\pi$$
$$600$$ −2.86564 −0.116989
$$601$$ 32.7460 1.33574 0.667869 0.744279i $$-0.267207\pi$$
0.667869 + 0.744279i $$0.267207\pi$$
$$602$$ −11.5492 −0.470712
$$603$$ −0.253176 −0.0103101
$$604$$ −5.94304 −0.241819
$$605$$ 0 0
$$606$$ −52.3556 −2.12680
$$607$$ 5.79559 0.235236 0.117618 0.993059i $$-0.462474\pi$$
0.117618 + 0.993059i $$0.462474\pi$$
$$608$$ 7.81494 0.316938
$$609$$ −8.58103 −0.347721
$$610$$ −7.82443 −0.316802
$$611$$ −29.0019 −1.17329
$$612$$ 12.1429 0.490848
$$613$$ 39.3606 1.58976 0.794879 0.606767i $$-0.207534\pi$$
0.794879 + 0.606767i $$0.207534\pi$$
$$614$$ 19.3442 0.780667
$$615$$ 19.7773 0.797498
$$616$$ 0 0
$$617$$ −16.3175 −0.656918 −0.328459 0.944518i $$-0.606529\pi$$
−0.328459 + 0.944518i $$0.606529\pi$$
$$618$$ 32.1946 1.29506
$$619$$ −44.7221 −1.79753 −0.898767 0.438427i $$-0.855536\pi$$
−0.898767 + 0.438427i $$0.855536\pi$$
$$620$$ 5.08533 0.204232
$$621$$ −50.6079 −2.03083
$$622$$ 1.82729 0.0732675
$$623$$ 18.5268 0.742263
$$624$$ −16.1527 −0.646626
$$625$$ 1.00000 0.0400000
$$626$$ −9.83108 −0.392929
$$627$$ 0 0
$$628$$ −16.7947 −0.670183
$$629$$ −3.23106 −0.128831
$$630$$ −5.21187 −0.207646
$$631$$ −30.1105 −1.19868 −0.599340 0.800495i $$-0.704570\pi$$
−0.599340 + 0.800495i $$0.704570\pi$$
$$632$$ −1.05291 −0.0418825
$$633$$ 75.7951 3.01258
$$634$$ −13.8545 −0.550232
$$635$$ −6.58727 −0.261408
$$636$$ −1.66979 −0.0662115
$$637$$ 5.63670 0.223334
$$638$$ 0 0
$$639$$ 23.2116 0.918237
$$640$$ 1.00000 0.0395285
$$641$$ −0.937781 −0.0370401 −0.0185201 0.999828i $$-0.505895\pi$$
−0.0185201 + 0.999828i $$0.505895\pi$$
$$642$$ 14.0175 0.553225
$$643$$ 44.9904 1.77425 0.887125 0.461529i $$-0.152699\pi$$
0.887125 + 0.461529i $$0.152699\pi$$
$$644$$ −7.98434 −0.314627
$$645$$ −33.0959 −1.30315
$$646$$ 18.2077 0.716373
$$647$$ −32.9160 −1.29406 −0.647032 0.762463i $$-0.723990\pi$$
−0.647032 + 0.762463i $$0.723990\pi$$
$$648$$ 2.52795 0.0993072
$$649$$ 0 0
$$650$$ 5.63670 0.221089
$$651$$ 14.5727 0.571149
$$652$$ 2.43041 0.0951821
$$653$$ −3.32100 −0.129961 −0.0649804 0.997887i $$-0.520698\pi$$
−0.0649804 + 0.997887i $$0.520698\pi$$
$$654$$ 17.9516 0.701961
$$655$$ −1.20821 −0.0472086
$$656$$ −6.90154 −0.269460
$$657$$ 1.75543 0.0684857
$$658$$ 5.14520 0.200581
$$659$$ 20.5642 0.801069 0.400535 0.916282i $$-0.368824\pi$$
0.400535 + 0.916282i $$0.368824\pi$$
$$660$$ 0 0
$$661$$ −43.9387 −1.70902 −0.854508 0.519438i $$-0.826141\pi$$
−0.854508 + 0.519438i $$0.826141\pi$$
$$662$$ 8.02084 0.311739
$$663$$ −37.6335 −1.46157
$$664$$ −10.8315 −0.420343
$$665$$ −7.81494 −0.303050
$$666$$ −7.22784 −0.280073
$$667$$ −23.9088 −0.925752
$$668$$ 11.0306 0.426786
$$669$$ −70.4640 −2.72430
$$670$$ −0.0485769 −0.00187669
$$671$$ 0 0
$$672$$ 2.86564 0.110544
$$673$$ 0.623385 0.0240297 0.0120149 0.999928i $$-0.496175\pi$$
0.0120149 + 0.999928i $$0.496175\pi$$
$$674$$ 11.6060 0.447045
$$675$$ −6.33840 −0.243965
$$676$$ 18.7723 0.722013
$$677$$ 30.6611 1.17840 0.589200 0.807987i $$-0.299443\pi$$
0.589200 + 0.807987i $$0.299443\pi$$
$$678$$ −29.1294 −1.11871
$$679$$ 5.33180 0.204616
$$680$$ 2.32986 0.0893460
$$681$$ −16.0258 −0.614112
$$682$$ 0 0
$$683$$ −4.08570 −0.156335 −0.0781676 0.996940i $$-0.524907\pi$$
−0.0781676 + 0.996940i $$0.524907\pi$$
$$684$$ 40.7304 1.55737
$$685$$ 14.0829 0.538079
$$686$$ −1.00000 −0.0381802
$$687$$ 23.2214 0.885951
$$688$$ 11.5492 0.440311
$$689$$ 3.28447 0.125128
$$690$$ −22.8802 −0.871035
$$691$$ −9.75385 −0.371054 −0.185527 0.982639i $$-0.559399\pi$$
−0.185527 + 0.982639i $$0.559399\pi$$
$$692$$ −14.6947 −0.558607
$$693$$ 0 0
$$694$$ 26.4301 1.00327
$$695$$ −2.74290 −0.104044
$$696$$ 8.58103 0.325263
$$697$$ −16.0796 −0.609059
$$698$$ −27.3777 −1.03626
$$699$$ 34.3323 1.29857
$$700$$ −1.00000 −0.0377964
$$701$$ 8.34928 0.315348 0.157674 0.987491i $$-0.449600\pi$$
0.157674 + 0.987491i $$0.449600\pi$$
$$702$$ −35.7276 −1.34845
$$703$$ −10.8378 −0.408755
$$704$$ 0 0
$$705$$ 14.7443 0.555301
$$706$$ 21.3772 0.804541
$$707$$ −18.2701 −0.687119
$$708$$ −39.6294 −1.48937
$$709$$ 41.6502 1.56421 0.782103 0.623149i $$-0.214147\pi$$
0.782103 + 0.623149i $$0.214147\pi$$
$$710$$ 4.45361 0.167141
$$711$$ −5.48763 −0.205802
$$712$$ −18.5268 −0.694323
$$713$$ 40.6030 1.52059
$$714$$ 6.67652 0.249863
$$715$$ 0 0
$$716$$ 4.51392 0.168693
$$717$$ −18.5295 −0.691998
$$718$$ 10.8904 0.406427
$$719$$ 27.1930 1.01413 0.507064 0.861909i $$-0.330731\pi$$
0.507064 + 0.861909i $$0.330731\pi$$
$$720$$ 5.21187 0.194235
$$721$$ 11.2347 0.418403
$$722$$ 42.0733 1.56581
$$723$$ 58.2986 2.16815
$$724$$ −10.1770 −0.378225
$$725$$ −2.99446 −0.111211
$$726$$ 0 0
$$727$$ 26.4443 0.980766 0.490383 0.871507i $$-0.336857\pi$$
0.490383 + 0.871507i $$0.336857\pi$$
$$728$$ −5.63670 −0.208910
$$729$$ −41.3153 −1.53020
$$730$$ 0.336814 0.0124660
$$731$$ 26.9081 0.995232
$$732$$ 22.4220 0.828740
$$733$$ −25.1965 −0.930653 −0.465326 0.885139i $$-0.654063\pi$$
−0.465326 + 0.885139i $$0.654063\pi$$
$$734$$ 11.3108 0.417487
$$735$$ −2.86564 −0.105701
$$736$$ 7.98434 0.294307
$$737$$ 0 0
$$738$$ −35.9699 −1.32407
$$739$$ −5.90682 −0.217286 −0.108643 0.994081i $$-0.534651\pi$$
−0.108643 + 0.994081i $$0.534651\pi$$
$$740$$ −1.38681 −0.0509800
$$741$$ −126.233 −4.63727
$$742$$ −0.582695 −0.0213914
$$743$$ −19.5944 −0.718847 −0.359424 0.933175i $$-0.617027\pi$$
−0.359424 + 0.933175i $$0.617027\pi$$
$$744$$ −14.5727 −0.534261
$$745$$ 17.1642 0.628849
$$746$$ −11.6658 −0.427116
$$747$$ −56.4523 −2.06548
$$748$$ 0 0
$$749$$ 4.89157 0.178734
$$750$$ −2.86564 −0.104638
$$751$$ 0.465285 0.0169785 0.00848925 0.999964i $$-0.497298\pi$$
0.00848925 + 0.999964i $$0.497298\pi$$
$$752$$ −5.14520 −0.187626
$$753$$ −36.0619 −1.31417
$$754$$ −16.8789 −0.614692
$$755$$ −5.94304 −0.216289
$$756$$ 6.33840 0.230525
$$757$$ −6.85681 −0.249215 −0.124607 0.992206i $$-0.539767\pi$$
−0.124607 + 0.992206i $$0.539767\pi$$
$$758$$ 0.617088 0.0224136
$$759$$ 0 0
$$760$$ 7.81494 0.283478
$$761$$ −41.5119 −1.50481 −0.752403 0.658703i $$-0.771105\pi$$
−0.752403 + 0.658703i $$0.771105\pi$$
$$762$$ 18.8767 0.683831
$$763$$ 6.26442 0.226787
$$764$$ 11.9914 0.433834
$$765$$ 12.1429 0.439028
$$766$$ −26.4471 −0.955573
$$767$$ 77.9510 2.81465
$$768$$ −2.86564 −0.103405
$$769$$ 17.8632 0.644164 0.322082 0.946712i $$-0.395617\pi$$
0.322082 + 0.946712i $$0.395617\pi$$
$$770$$ 0 0
$$771$$ −9.91495 −0.357078
$$772$$ 17.8748 0.643330
$$773$$ 9.91463 0.356604 0.178302 0.983976i $$-0.442940\pi$$
0.178302 + 0.983976i $$0.442940\pi$$
$$774$$ 60.1931 2.16360
$$775$$ 5.08533 0.182670
$$776$$ −5.33180 −0.191400
$$777$$ −3.97408 −0.142569
$$778$$ 1.60446 0.0575226
$$779$$ −53.9352 −1.93243
$$780$$ −16.1527 −0.578360
$$781$$ 0 0
$$782$$ 18.6024 0.665220
$$783$$ 18.9801 0.678293
$$784$$ 1.00000 0.0357143
$$785$$ −16.7947 −0.599430
$$786$$ 3.46228 0.123496
$$787$$ 0.812618 0.0289667 0.0144834 0.999895i $$-0.495390\pi$$
0.0144834 + 0.999895i $$0.495390\pi$$
$$788$$ 11.7871 0.419897
$$789$$ −22.9763 −0.817979
$$790$$ −1.05291 −0.0374609
$$791$$ −10.1651 −0.361429
$$792$$ 0 0
$$793$$ −44.1039 −1.56618
$$794$$ 5.28819 0.187671
$$795$$ −1.66979 −0.0592214
$$796$$ −13.9342 −0.493884
$$797$$ 32.3331 1.14530 0.572649 0.819800i $$-0.305916\pi$$
0.572649 + 0.819800i $$0.305916\pi$$
$$798$$ 22.3948 0.792766
$$799$$ −11.9876 −0.424090
$$800$$ 1.00000 0.0353553
$$801$$ −96.5595 −3.41176
$$802$$ −13.6059 −0.480441
$$803$$ 0 0
$$804$$ 0.139204 0.00490934
$$805$$ −7.98434 −0.281411
$$806$$ 28.6645 1.00966
$$807$$ 53.2005 1.87275
$$808$$ 18.2701 0.642741
$$809$$ 24.5501 0.863135 0.431568 0.902081i $$-0.357961\pi$$
0.431568 + 0.902081i $$0.357961\pi$$
$$810$$ 2.52795 0.0888230
$$811$$ −41.3394 −1.45162 −0.725812 0.687893i $$-0.758536\pi$$
−0.725812 + 0.687893i $$0.758536\pi$$
$$812$$ 2.99446 0.105085
$$813$$ 72.2287 2.53317
$$814$$ 0 0
$$815$$ 2.43041 0.0851335
$$816$$ −6.67652 −0.233725
$$817$$ 90.2567 3.15768
$$818$$ 3.40230 0.118959
$$819$$ −29.3777 −1.02654
$$820$$ −6.90154 −0.241012
$$821$$ −16.0197 −0.559090 −0.279545 0.960133i $$-0.590184\pi$$
−0.279545 + 0.960133i $$0.590184\pi$$
$$822$$ −40.3564 −1.40759
$$823$$ 19.3583 0.674787 0.337394 0.941364i $$-0.390455\pi$$
0.337394 + 0.941364i $$0.390455\pi$$
$$824$$ −11.2347 −0.391380
$$825$$ 0 0
$$826$$ −13.8292 −0.481179
$$827$$ −43.1757 −1.50136 −0.750682 0.660663i $$-0.770275\pi$$
−0.750682 + 0.660663i $$0.770275\pi$$
$$828$$ 41.6133 1.44616
$$829$$ −38.5839 −1.34007 −0.670037 0.742327i $$-0.733722\pi$$
−0.670037 + 0.742327i $$0.733722\pi$$
$$830$$ −10.8315 −0.375966
$$831$$ −35.0412 −1.21557
$$832$$ 5.63670 0.195417
$$833$$ 2.32986 0.0807248
$$834$$ 7.86016 0.272175
$$835$$ 11.0306 0.381729
$$836$$ 0 0
$$837$$ −32.2329 −1.11413
$$838$$ −39.5263 −1.36541
$$839$$ 9.31984 0.321757 0.160878 0.986974i $$-0.448567\pi$$
0.160878 + 0.986974i $$0.448567\pi$$
$$840$$ 2.86564 0.0988738
$$841$$ −20.0332 −0.690800
$$842$$ −15.6617 −0.539738
$$843$$ 76.2071 2.62471
$$844$$ −26.4497 −0.910435
$$845$$ 18.7723 0.645788
$$846$$ −26.8161 −0.921956
$$847$$ 0 0
$$848$$ 0.582695 0.0200098
$$849$$ −39.3223 −1.34954
$$850$$ 2.32986 0.0799135
$$851$$ −11.0727 −0.379568
$$852$$ −12.7624 −0.437233
$$853$$ −9.65688 −0.330645 −0.165323 0.986240i $$-0.552867\pi$$
−0.165323 + 0.986240i $$0.552867\pi$$
$$854$$ 7.82443 0.267747
$$855$$ 40.7304 1.39295
$$856$$ −4.89157 −0.167190
$$857$$ −7.88344 −0.269293 −0.134647 0.990894i $$-0.542990\pi$$
−0.134647 + 0.990894i $$0.542990\pi$$
$$858$$ 0 0
$$859$$ 9.00176 0.307136 0.153568 0.988138i $$-0.450924\pi$$
0.153568 + 0.988138i $$0.450924\pi$$
$$860$$ 11.5492 0.393826
$$861$$ −19.7773 −0.674009
$$862$$ −5.63150 −0.191810
$$863$$ −14.5890 −0.496616 −0.248308 0.968681i $$-0.579874\pi$$
−0.248308 + 0.968681i $$0.579874\pi$$
$$864$$ −6.33840 −0.215637
$$865$$ −14.6947 −0.499633
$$866$$ −34.5223 −1.17311
$$867$$ 33.1604 1.12619
$$868$$ −5.08533 −0.172607
$$869$$ 0 0
$$870$$ 8.58103 0.290924
$$871$$ −0.273813 −0.00927781
$$872$$ −6.26442 −0.212140
$$873$$ −27.7886 −0.940502
$$874$$ 62.3972 2.11062
$$875$$ −1.00000 −0.0338062
$$876$$ −0.965185 −0.0326106
$$877$$ −17.0812 −0.576791 −0.288396 0.957511i $$-0.593122\pi$$
−0.288396 + 0.957511i $$0.593122\pi$$
$$878$$ −30.0533 −1.01425
$$879$$ 2.04739 0.0690568
$$880$$ 0 0
$$881$$ −30.9541 −1.04287 −0.521435 0.853291i $$-0.674603\pi$$
−0.521435 + 0.853291i $$0.674603\pi$$
$$882$$ 5.21187 0.175493
$$883$$ −13.5966 −0.457561 −0.228781 0.973478i $$-0.573474\pi$$
−0.228781 + 0.973478i $$0.573474\pi$$
$$884$$ 13.1327 0.441700
$$885$$ −39.6294 −1.33213
$$886$$ −32.0651 −1.07725
$$887$$ −17.1901 −0.577188 −0.288594 0.957452i $$-0.593188\pi$$
−0.288594 + 0.957452i $$0.593188\pi$$
$$888$$ 3.97408 0.133361
$$889$$ 6.58727 0.220930
$$890$$ −18.5268 −0.621021
$$891$$ 0 0
$$892$$ 24.5893 0.823311
$$893$$ −40.2094 −1.34556
$$894$$ −49.1865 −1.64504
$$895$$ 4.51392 0.150884
$$896$$ −1.00000 −0.0334077
$$897$$ −128.969 −4.30614
$$898$$ −9.27706 −0.309579
$$899$$ −15.2278 −0.507876
$$900$$ 5.21187 0.173729
$$901$$ 1.35760 0.0452281
$$902$$ 0 0
$$903$$ 33.0959 1.10136
$$904$$ 10.1651 0.338086
$$905$$ −10.1770 −0.338295
$$906$$ 17.0306 0.565803
$$907$$ −23.5921 −0.783363 −0.391682 0.920101i $$-0.628107\pi$$
−0.391682 + 0.920101i $$0.628107\pi$$
$$908$$ 5.59242 0.185591
$$909$$ 95.2215 3.15830
$$910$$ −5.63670 −0.186855
$$911$$ 50.0697 1.65888 0.829441 0.558594i $$-0.188659\pi$$
0.829441 + 0.558594i $$0.188659\pi$$
$$912$$ −22.3948 −0.741565
$$913$$ 0 0
$$914$$ 20.0782 0.664127
$$915$$ 22.4220 0.741247
$$916$$ −8.10340 −0.267744
$$917$$ 1.20821 0.0398985
$$918$$ −14.7676 −0.487403
$$919$$ 13.1628 0.434200 0.217100 0.976149i $$-0.430340\pi$$
0.217100 + 0.976149i $$0.430340\pi$$
$$920$$ 7.98434 0.263236
$$921$$ −55.4333 −1.82659
$$922$$ −14.6490 −0.482439
$$923$$ 25.1036 0.826296
$$924$$ 0 0
$$925$$ −1.38681 −0.0455979
$$926$$ 11.0973 0.364681
$$927$$ −58.5539 −1.92316
$$928$$ −2.99446 −0.0982979
$$929$$ −9.87262 −0.323910 −0.161955 0.986798i $$-0.551780\pi$$
−0.161955 + 0.986798i $$0.551780\pi$$
$$930$$ −14.5727 −0.477858
$$931$$ 7.81494 0.256124
$$932$$ −11.9807 −0.392441
$$933$$ −5.23634 −0.171430
$$934$$ 24.6923 0.807956
$$935$$ 0 0
$$936$$ 29.3777 0.960240
$$937$$ −4.81983 −0.157457 −0.0787285 0.996896i $$-0.525086\pi$$
−0.0787285 + 0.996896i $$0.525086\pi$$
$$938$$ 0.0485769 0.00158609
$$939$$ 28.1723 0.919368
$$940$$ −5.14520 −0.167818
$$941$$ 57.3904 1.87087 0.935437 0.353493i $$-0.115006\pi$$
0.935437 + 0.353493i $$0.115006\pi$$
$$942$$ 48.1276 1.56808
$$943$$ −55.1043 −1.79444
$$944$$ 13.8292 0.450102
$$945$$ 6.33840 0.206188
$$946$$ 0 0
$$947$$ −4.22221 −0.137203 −0.0686017 0.997644i $$-0.521854\pi$$
−0.0686017 + 0.997644i $$0.521854\pi$$
$$948$$ 3.01726 0.0979959
$$949$$ 1.89852 0.0616284
$$950$$ 7.81494 0.253550
$$951$$ 39.7019 1.28742
$$952$$ −2.32986 −0.0755111
$$953$$ 5.10500 0.165367 0.0826836 0.996576i $$-0.473651\pi$$
0.0826836 + 0.996576i $$0.473651\pi$$
$$954$$ 3.03693 0.0983241
$$955$$ 11.9914 0.388033
$$956$$ 6.46612 0.209129
$$957$$ 0 0
$$958$$ 31.2114 1.00839
$$959$$ −14.0829 −0.454759
$$960$$ −2.86564 −0.0924880
$$961$$ −5.13941 −0.165787
$$962$$ −7.81700 −0.252030
$$963$$ −25.4942 −0.821539
$$964$$ −20.3441 −0.655238
$$965$$ 17.8748 0.575412
$$966$$ 22.8802 0.736159
$$967$$ −61.6521 −1.98260 −0.991298 0.131633i $$-0.957978\pi$$
−0.991298 + 0.131633i $$0.957978\pi$$
$$968$$ 0 0
$$969$$ −52.1766 −1.67616
$$970$$ −5.33180 −0.171194
$$971$$ −26.0709 −0.836656 −0.418328 0.908296i $$-0.637384\pi$$
−0.418328 + 0.908296i $$0.637384\pi$$
$$972$$ 11.7710 0.377556
$$973$$ 2.74290 0.0879334
$$974$$ 3.02777 0.0970160
$$975$$ −16.1527 −0.517301
$$976$$ −7.82443 −0.250454
$$977$$ 19.0437 0.609261 0.304631 0.952471i $$-0.401467\pi$$
0.304631 + 0.952471i $$0.401467\pi$$
$$978$$ −6.96466 −0.222705
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ −32.6493 −1.04241
$$982$$ 14.8069 0.472509
$$983$$ −19.3347 −0.616681 −0.308340 0.951276i $$-0.599774\pi$$
−0.308340 + 0.951276i $$0.599774\pi$$
$$984$$ 19.7773 0.630478
$$985$$ 11.7871 0.375568
$$986$$ −6.97666 −0.222182
$$987$$ −14.7443 −0.469315
$$988$$ 44.0504 1.40143
$$989$$ 92.2131 2.93221
$$990$$ 0 0
$$991$$ −28.4567 −0.903958 −0.451979 0.892029i $$-0.649282\pi$$
−0.451979 + 0.892029i $$0.649282\pi$$
$$992$$ 5.08533 0.161459
$$993$$ −22.9848 −0.729401
$$994$$ −4.45361 −0.141260
$$995$$ −13.9342 −0.441743
$$996$$ 31.0391 0.983511
$$997$$ −14.1144 −0.447008 −0.223504 0.974703i $$-0.571750\pi$$
−0.223504 + 0.974703i $$0.571750\pi$$
$$998$$ −6.99445 −0.221406
$$999$$ 8.79013 0.278107
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8470.2.a.dc.1.1 6
11.7 odd 10 770.2.n.j.71.3 12
11.8 odd 10 770.2.n.j.141.3 yes 12
11.10 odd 2 8470.2.a.cw.1.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.j.71.3 12 11.7 odd 10
770.2.n.j.141.3 yes 12 11.8 odd 10
8470.2.a.cw.1.1 6 11.10 odd 2
8470.2.a.dc.1.1 6 1.1 even 1 trivial